2 FORMAL RULES OF DERIVATION 153

By assumption, the mapping s-+f(xQ + s) -/(*0) is a homeomorphism
of a neighborhood V of 0 in E onto a neighborhood W of 0 in F, and
the inverse homeomorphism if t-*g(yQ + t)-g(yQ). By assumption, the
linear mapping fr(xQ) of E onto F has an inverse u which is continuous,
hence (5.5.1) there is c> 0 such that \\u(t)\\ <c\\t\\ for any r eF. Given
any e such that 0<e< l/2c, there is an r > 0 such that, if we write
/(*o + -0 ~/Oo) =/'(*(>) * s + 0iO), the relation ||,s|| ^ rimplies H^COH ^ 8||s||.
Let rr now be a number such that the ball \\t\\ < r' is contained in W and
that its image by the mapping t -> g(yQ + t) - g(y0) is contained in the
ball \\s\\ ^ r. Let z = #(j;0 + 0 - g(yQ)i by definition, for \\t\\ < r',
this equation implies t=f(xQ +z)—f(x0) and as ||z|| ^r, we can write
t = ff(x0) - z + ^O), with ||0i(z)|| < e||z||. From that relation we deduce

u - t = u • (/'Oo) * z) + w • ox(z) = z + u - ot(z)

by definition of w, and moreover ||w • ^(z)!! < cH^^z)!] < cŁ\\z\\ ^ i||z||,
hence \\u • r|| > ||z|| - ± ||z|| = i ||z||; therefore ||z|| < 2||i/ • f|| < 2c\\t\\, and
finally Hwo^U ^c&\\z\\ <2c2«p||. We have therefore proved that the
relation ||f || < rr implies \\g(y0 + t) — g(y0} — u • t \\ < 2c2e ||f ||, and as e is
arbitrary, this completes the proof.

The result (8.2.3) can also be written (under the same assumptions)
(8.2.3.1) (/"1y(/(^o)) = (//(^o))"1.

PROBLEMS

1. Let E be a real prehilbert space. Show that in E the mapping x-> \\x\\ of E into R
is differentiable at every point x ^ 0 and that its derivative at such a point is the linear
mapping s-»(s\x)l\\x\\.

2. (a) In the space (c0) of Banach (Section 5.3, Problem 5) show that the norm x -* ||jc|]
is differentiable at a point x = (Łn) if and only if there is an index n0 such that
Ifn0l >lfn| for every n ^ nQ. Compute the derivative.

(b) In the space I1 of Banach (Section 5.7, Problem 1), show that the norm x-* \\x\\
is not differentiable at any point (use (8.1.1) and Problem l(c) of Section 5.7).

3. Let / be a differentiable real valued function defined in an open subset A of a Banach
space E.

(a) Show that if at a point XQ e A, /reaches a relative maximum (Section 3.9, Problem
6), then D/(*o) = 0.

(b) Suppose E is finite dimensional, A is relatively compact,/is defined and continuous
in A, and equal to 0 in the boundary of A. Show that there exists a point XQ e A
where D/(x0) = 0 (" Rolle's theorem "; use (a) and (3.17.10)).all purposes of